Convergence of Schrodinger operators

TL;DR

This paper demonstrates that certain Schrödinger operators with additional higher-order terms converge to standard Schrödinger operators as parameters tend to zero, enabling efficient numerical eigenvalue computations.

Contribution

It establishes convergence of a family of Schrödinger-type operators with measure potentials to the standard Schrödinger operator, including a method for numerical eigenvalue approximation.

Findings

Operators with higher-order derivatives converge to standard Schrödinger operators as C→0.

Finite measures in low dimensions can be approximated by point measures.

Numerical methods successfully compute eigenvalues for measure-based Schrödinger operators.

Abstract

For a large family of real-valued Radon measures m on R^d, including the Kato class, the operators -\Delta + C^2 \Delta^2 + m tend to the Schrodinger operator -\Delta +m in the norm resolvent sense as C tends to zero. If the measure is moreover finite and the dimension smaller than four, the former operator can be approximated by a sequence of operators with point measures in the norm resolvent sense. The combination of both convergence results leads to an efficient method for the numerical computation of the eigenvalues in the discrete spectrum and corresponding eigenfunctions of Schrodinger operators. We illustrate the approximation by numerical calculations of eigenvalues for one simple example of measure m.